The α-divergences associated with a pair of strictly comparable quasi-arithmetic means
英文摘要：
We generalize the family of $α$-divergences using a pair of strictly comparable weighted means. In particular, we obtain the $1$-divergence in the limit case $α\rightarrow 1$ (a generalization of the Kullback-Leibler divergence) and the $0$-divergence in the limit case $α\rightarrow 0$ (a generalization of the reverse Kullback-Leibler divergence). We state the condition for a pair of quasi-arithmetic means to be strictly comparable, and report the formula for the quasi-arithmetic $α$-divergences and its subfamily of bipower homogeneous $α$-divergences which belong to the Csisár's $f$-divergences. Finally, we show that these generalized quasi-arithmetic $1$-divergences and $0$-divergences can be decomposed as the sum of generalized cross-entropies minus entropies, and rewritten as conformal Bregman divergences using monotone embeddings.
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